Dynamical stability of infinite homogeneous self-gravitating systems and plasmas: application of the Nyquist method

2013

Eur. Phys. J. Plus

We solve the initial value problem for the linearized mean field Kramers equation describing Brownian particles with long-range interactions in the N → +∞ limit. We show that the dielectric function can be expressed in terms of incomplete Gamma functions. The dielectric functions associated with the linearized Vlasov equation and with the linearized mean field Smoluchowski equation are recovered as special cases corresponding to the no friction limit or to the strong friction limit respectively. Although the stability of the Maxwell-Boltzmann distribution is independent on the friction parameter, the evolution of the perturbation depends on it in a non-trivial manner. For illustration, we apply our results to self-gravitating systems, plasmas, and to the attractive and repulsive BMF models.

Section: Introductionmentioning

confidence: 99%

Initial value problem for the linearized mean field Kramers equation with long-range interactions

2013

Eur. Phys. J. Plus

“…Remark: If we consider ultralight bosons with mass m = 2.19 × 10 −22 eV/c 2 and negative scattering length a s = −1.11×10 −62 fm (see Sec. VIII E 2) in the ultrarelativistic limit where ρ i = 1.15 × 10 −9 g/m 3 , we obtain σ = 2.87 × 10 7 1, λ J = 0.159 pc and M J = 1.78 × 10 4 M . These Jeans scales are much smaller than the Hubble scales λ H = 542 pc and M H = 5.66 × 10 15 M implying that the Jeans instability can take place.…”

Section: The Nongravitational Limitmentioning

confidence: 63%

“…Even though today processes of stellar formation and galaxy formation are known with precision, the Jeans theory remains a good first approximation and has some pedagogical virtue. We refer to [7] for a review of the Jeans instability problem for collisional gaseous systems described by the Euler-Poisson equations and for collisionless stellar systems described by the Vlasov-Poisson equations.…”

Section: Introduction: a Brief Reviewmentioning

confidence: 99%

“…2 Secondly, the results of Eqs. (3) and (4) suggest 2 See the Introduction of [7] for a more detailed discussion of the that there is no upper limit to the Jeans instability since the maximum growth rate is achieved for λ * → +∞. Thirdly, as noted by Weinberg [9], the Jeans analysis is not rigorously applicable to the formation of galaxies in an expanding Universe because Jeans assumed a static medium whereas the rate of expansion of the Universe with a scale factor a(t) is given by the Hubble parameter H =ȧ a = 8πGρ 3 1/2 (6) which is of the same order as the maximum growth rate given by Eq.…”

Section: Introduction: a Brief Reviewmentioning

confidence: 99%

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Jeans-type instability of a complex self-interacting scalar field in general relativity

Suárez

2018

Phys. Rev. D

We study the gravitational instability of a general relativistic complex scalar field with a quartic self-interaction in an infinite homogeneous static background. This quantum relativistic Jeans problem provides a simplified framework to study the formation of the large-scale structures of the Universe in the case where dark matter is made of a scalar field. The scalar field may represent the wave function of a relativistic self-gravitating Bose-Einstein condensate. Exact analytical expressions for the dispersion relation and Jeans length λJ are obtained from a hydrodynamical representation of the Klein-Gordon-Einstein equations [Suárez and Chavanis, Phys. Rev. D 92, 023510 (2015)]. We compare our results with those previously obtained with a simplified relativistic model [Khlopov, Malomed and Y. B. Zeldovich, Mon. Not. R. astr. Soc. 215, 575 (1985)]. When relativistic effects are fully accounted for, we find that the perturbations are stabilized at very large scales of the order of the Hubble length λH . Numerical applications are made for ultralight bosons without self-interaction (fuzzy dark matter), for bosons with a repulsive self-interaction, and for bosons with an attractive self-interaction (QCD axions and ultralight axions). We show that the Jeans instability is inhibited in the ultrarelativistic regime (early Universe and radiationlike era) because the Jeans length is of the order of the Hubble length (λJ ∼ λH ), except when the self-interaction of the bosons is attractive. By contrast, structure formation can take place in the nonrelativistic regime (matterlike era) for λJ ≤ λ ≤ λH . Since the scalar field has a nonzero Jeans length due to its quantum nature (Heisenberg uncertainty principle or quantum pressure due to the self-interaction of the bosons), it appears that the wave properties of the scalar field can stabilize gravitational collapse at small scales, providing halo cores and suppressing small-scale linear power. This may solve the CDM crisis such as the cusp problem and the missing satellite problem. We also compare our results with those obtained in the case where dark matter is made of fermions instead of bosons. PACS numbers: 95.35.+d, 98.80.Jk, 95.30.Sf

“…38 There is an exception. In the case of an infinite and homogeneous medium, collisionless stellar systems (Vlasov) and self-gravitating fluids (Euler) behave in the same way with respect to the Jeans instability in the sense that they lead to the same criterion for instability [313]. 39 To make the correspondance with Appendix D 2 we just need to replace k B T (r) by e −ν(r)/2 .…”

Section: General Relativity: Vlasov-einstein Equationsmentioning

confidence: 99%

Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

2020

Eur. Phys. J. Plus

We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi-Dirac entropy for fermions. The maximization of entropy at fixed mass-energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium and the Tolman-Klein relations. Our paper provides all the necessary equations that are needed to construct the caloric curves of selfgravitating fermions in general relativity as done in recent works. We consider the nonrelativistic limit c → +∞ and recover the equations obtained within the framework of Newtonian gravity. We also discuss the inequivalence of statistical ensembles as well as the relation between the dynamical and thermodynamical stability of self-gravitating systems in Newtonian gravity and general relativity.PACS numbers: 04.40. Dg, 05.70.Fh, 95.30.Sf, 95.35.+d I. INTRODUCTIONSelf-gravitating fermions play an important role in different areas of astrophysics. They appeared in the context of white dwarfs, neutron stars and dark matter halos where the fermions are electrons, neutrons and massive neutrinos respectively. We start by a brief history of the subject giving an exhaustive list of references. 1 Soon after the discovery of the quantum statistics by Fermi [3,4] and Dirac [5] in 1926, Fowler [6] used this "new thermodynamics" to solve the puzzle of the extreme high density of white dwarfs, which could not be explained by classical physics [7]. He understood that white dwarfs owe their stability to the quantum pressure of the degenerate electron gas resulting from the Pauli exclusion principle [8]. He considered a completely degenerate electron gas at T = 0 based on the fact that the temperature in white dwarfs is much smaller than the Fermi temperature (T ≪ T F ). He also used Newtonian gravity which is a very good approximation to describe white dwarfs in general. The first models [9-11] of white dwarfs were based on the nonrelativistic equation of state of a Fermi gas and provided the corresponding mass-radius relation. Stoner [9] developed an analytical approach based on a uniform density approximation for the star. Chandrasekhar [11] derived the exact mass-radius relation of nonrelativistic white dwarfs by applying the theory of polytropes of index n = 3/2 [12]. It was then realized that special relativity must be taken into account at high densities. When the relativistic equation of state is employed it was found that white dwarfs can exist only below a maximum mass M max = 1.42 M ⊙ [13-25], now known as the Chandrasekhar limiting mass. Frenkel [13] was the first to mention that relativistic effects become important when the mass of white dwarfs becomes larger than the solar mass but he did not envision the existence of an upper mass limit. The maxi...